• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2008.04a
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• pp.125-126
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• 2008

A near-ring N is called strongly reduced if, for a ${\epsilon}$ N, $a^2\;{\epsilon}\;N_c$ implies a ${\epsilon}\;N_c$, where $N_c$ denotes the constant part of N. We investigate some properties of strongly reduced near-rings and apply those to the study of left strongly regular near-rings. Finally we classify some reduced, and strongly reduced near-rings.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.1-7
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• 2008

A near-ring N is said to be strongly reduced if, for a ${\in}$ N, $a^2{\in}N_c$ implies $a{\in}N_c$, where $N_c$ denotes the constant part of N. We investigate some properties of strongly reduced near-rings and apply those to the study of left strongly regular near-rings. Finally we classify all reduced and strongly reduced near-rings of order ${\leq}$ 7 using the description given in J. R. Clay [1].

• Korean Journal of Mathematics
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• v.29 no.3
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• pp.483-492
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• 2021

This article concerns the property that for any element a in a ring, if a²ⁿ = aⁿ for some n ≥ 2 then a² = a. The class of rings with this property is large, but there also exist many kinds of rings without that, for example, rings of characteristic ≠2 and finite fields of characteristic ≥ 3. Rings with such a property is called reduced-over-idempotent. The study of reduced-over-idempotent rings is based on the fact that the characteristic is 2 and every nonzero non-identity element generates an infinite multiplicative semigroup without identity. It is proved that the reduced-over-idempotent property pass to polynomial rings, and we provide power series rings with a partial affirmative argument. It is also proved that every finitely generated subring of a locally finite reduced-over-idempotent ring is isomorphic to a finite direct product of copies of the prime field {0, 1}. A method to construct reduced-over-idempotent fields is also provided.

• East Asian mathematical journal
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• v.22 no.1
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• pp.61-69
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• 2006

A near-ring N is reduced if, for $a{\in}N,\;a^2=0$ implies a=0, and N is left strongly regular if for all $a{\in}N$ there exists $x{\in}N$ such that $a=xa^2$. Mason introduced this notion and characterized left strongly regular zero-symmetric unital near-rings. Several authors ([2], [5], [7]) studied these properties in near-rings. Reddy and Murty extended some results in Mason to the non-zero symmetric case. In this paper, we will define a concept of strong reducedness and investigate a relation between strongly reduced near-rings and left strongly regular near-rings.

• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.587-592
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• 2003

A near-ring N is called strongly reduced if, for $a{\in}N$, $a^2{\in}N_c$ implies $a{\in}N_c$, where $N_c$ denotes the constant part of N. We investigate some properties of strongly reduced near-rings and apply those to the study of left strongly regular near-rings. Finally we classify all reduced, and strongly reduced near-rings of order ${\leq}7$.

• The Pure and Applied Mathematics
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• v.17 no.2
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• pp.131-136
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• 2010

Throught this paper, we will investigate some properties of left regular and strongly reduced near-rings. Mason introduced the notion of left regularity and he characterized left regular zero-symmetric unital near-rings. Also, this concept have been studied by several authors. The purpose of this paper is to find some characterizations of the strong reducibility in near-rings, and the strong regularity in near-rings which are closely related with strongly reduced near-rings.

• East Asian mathematical journal
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• v.26 no.3
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• pp.397-401
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• 2010

We investigate some properties of strongly reduced near-rings and left regular near-rings, also show that a strongly reduced near-ring is reduced. Next, we will characterize that reducibility, strong reducibility and left regularity in near-rings.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.151-154
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• 2007

We study ${\sigma}$-derivations on reduced near-rings and extend certain results on derivations to ${\sigma}$-derivations with some conditions.

• The Pure and Applied Mathematics
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• v.10 no.4
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• pp.199-206
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• 2003

Let N be a right near-ring. N is said to be strongly reduced if, for $a\inN$, $a^2 \in N_{c}$ implies $a\;\in\;N_{c}$, or equivalently, for $a\inN$ and any positive integer n, $a^{n} \in N_{c}$ implies $a\;\in\;N_{c}$, where $N_{c}$ denotes the constant part of N. We will show that strong reducedness is equivalent to condition (ⅱ) of Reddy and Murty's property $(^{\ast})$ (cf. [Reddy & Murty: On strongly regular near-rings. Proc. Edinburgh Math. Soc. (2) 27 (1984), no. 1, 61-64]), and that condition (ⅰ) of Reddy and Murty's property $(^{\ast})$ follows from strong reducedness. Also, we will investigate some characterizations of strongly reduced near-rings and their properties. Using strong reducedness, we characterize left strongly regular near-rings and ($P_{0}$)-near-rings.

• Bulletin of the Korean Mathematical Society
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• v.46 no.1
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• pp.135-146
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• 2009

In the present note we study the properties of weak Armendariz rings, and the connections among weak Armendariz rings, Armendariz rings, reduced rings and IFP rings. We prove that a right Ore ring R is weak Armendariz if and only if so is Q, where Q is the classical right quotient ring of R. With the help of this result we can show that a semiprime right Goldie ring R is weak Armendariz if and only if R is Armendariz if and only if R is reduced if and only if R is IFP if and only if Q is a finite direct product of division rings, obtaining a simpler proof of Lee and Wong's result. In the process we construct a semiprime ring extension that is infinite dimensional, from given any semi prime ring. We next find more examples of weak Armendariz rings.